cp's OEIS Frontend

A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

%I A062991 #50 Sep 27 2024 09:24:34
%S A062991 1,2,-1,5,-6,2,14,-28,20,-5,42,-120,135,-70,14,132,-495,770,-616,252,
%T A062991 -42,429,-2002,4004,-4368,2730,-924,132,1430,-8008,19656,-27300,23100,
%U A062991 -11880,3432,-429,4862,-31824,92820,-157080,168300,-116688,51051,-12870,1430
%N A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).
%C A062991 The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) = Sum_{k=0..n} T(n,k)*x^k.
%C A062991 For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.
%C A062991 Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938.
%C A062991 The positive triangle is |T(n,k)| = binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1). - _Paul Barry_, May 11 2005
%H A062991 G. C. Greubel, <a href="/A062991/b062991.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A062991 C. A. Francisco, J. Mermin, and J. Schweig, <a href="http://www.math.okstate.edu/~jayjs/ppt.pdf">Catalan numbers, binary trees, and pointed pseudotriangulations</a>, 2013.
%H A062991 V. E. Hoggatt Jr. and Marjorie Bicknell-Johnson, <a href="https://fq.math.ca/Scanned/15-1/hoggatt3.pdf">Numerator Polynomial Coefficient Arrays for Catalan and Related Sequence Convolution Triangles</a>, The Fibonacci Quarterly 15 (1977) 30-34. [On p. 31, in the line n = 1, 14 is missing in S_1^4. - _Wolfdieter Lang_, Jan 20 2020 ]
%H A062991 Joseph T. Iosue, Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov, <a href="https://arxiv.org/abs/2209.06838">Page curves and typical entanglement in linear optics</a>, arXiv:2209.06838 [quant-ph], 2022.
%H A062991 A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, <a href="http://arxiv.org/abs/1407.1169">Diagonal unitary entangling gates and contradiagonal quantum states</a>, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
%H A062991 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A062991 T(n, k) = [x^k] N(2; n, x) with N(2; n, x) = (N(2; n-1, x) - A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) = 1.
%F A062991 T(n, k) = T(n-1, k+1) + (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); T(n, k) = (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.
%F A062991 O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. - _Peter Bala_, Jul 15 2012
%F A062991 From _G. C. Greubel_, Sep 27 2024: (Start)
%F A062991 Sum_{k=0..n} T(n, k) = A000012(n).
%F A062991 Sum_{k=0..n} (-1)^k*T(n, k) = A064062(n+1).
%F A062991 Sum_{k=0..floor(n/2)} T(n-k, k) = A000079(n).
%F A062991 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A071356(n). (End)
%e A062991 The triangle N2 = {a(n,k)} begins:
%e A062991 n\k      0       1      2       3       4       5      6       7     8     9
%e A062991 ----------------------------------------------------------------------------
%e A062991 0:       1
%e A062991 1:       2      -1
%e A062991 2:       5      -6      2
%e A062991 3:      14     -28     20      -5
%e A062991 4:      42    -120    135     -70      14
%e A062991 5:     132    -495    770    -616     252     -42
%e A062991 6:     429   -2002   4004   -4368    2730    -924    132
%e A062991 7:    1430   -8008  19656  -27300   23100  -11880   3432    -429
%e A062991 8:    4862  -31824  92820 -157080  168300 -116688  51051  -12870  1430
%e A062991 9:   16796 -125970 426360 -852720 1108536 -969969 570570 -217360 48620 -4862
%e A062991 ... formatted by _Wolfdieter Lang_, Jan 20 2020
%e A062991 N(2; 2, x)= 5 - 6*x + 2*x^2.
%t A062991 T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
%t A062991 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018 *)
%o A062991 (Magma)
%o A062991 A062991:= func< n,k | (-1)^k*Binomial(2*n+2,n-k)*Binomial(n+k,k)/(n+1) >;
%o A062991 [A062991(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2024
%o A062991 (SageMath)
%o A062991 def A062991(n,k): return (-1)^k*binomial(2*n+2,n-k)*binomial(n+k,k)/(n+1)
%o A062991 flatten([[A062991(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 27 2024
%Y A062991 Cf. A000108, A002694, A009766, A033184, A062992, A084938, A089434, A094385.
%Y A062991 For an unsigned version see Borel's triangle, A234950.
%Y A062991 Sums include: A000012 (row), A000079 (diagonal), A064062 (signed row), A071356 (signed diagonal).
%K A062991 sign,easy,tabl
%O A062991 0,2
%A A062991 _Wolfdieter Lang_, Jul 12 2001